We have already seen symmetric polynomials and some of their applications in an earlier article. Let us delve into this a little more deeply. Consider the ring of integer polynomials. The symmetric group acts on it by permuting the variables; specifically, takes:

For example, takes to . Denote the ring of symmetric polynomials by:

This is a homogeneous ring, written as a direct sum:

where is the additive group of homogeneous polynomials of degree . For example, when , we have (upon renaming the variables to x, y, z):

Let us fix n, the number of variables.

Monomial Basis

Clearly, each is a free additive abelian group. For example, for , a basis is given by:

More generally, recall that a partition of is a sequence of non-negative integers:

Two partitions of are considered the same if we can obtain one from the other by appending or dropping zeros. E.g. (3, 2, 1, 0) and (3, 2, 1) are identical partitions of 6.

Definition. Given a partition of , let be the symmetric polynomial obtained by summing all terms of the form:

For example when , we have , and .

The following result is clear.

Theorem. The additive group is free with basis given by , for all partitions of into at most parts.

Let us consider an example where . A basis of is given by:

Note that for brevity of notation, we write instead of . However, if the subscripts involve variables, we will include the commas for clarity.

Elementary Symmetric Polynomials

Recall that the elementary symmetric polynomials in x, y, z are given by x+y+z, xy+yz+zx and xyz. Generalizing this, we define:

Furthermore, given a partition of , we define:

assuming . Note that since , we can take as many terms as we want by increasing without affecting For example, when we have:

Now since (here, where denotes the sum of all ), it is natural to ask the following question.

Task. Express the symmetric polynomial as a -linear combination of the monomial polynomials for various

In our above example, expanding with WolframAlpha gives us: In the remaining of this article, we will describe a more systematic method of computation.

Expanding the e‘s

Here is our main result.

Theorem. Suppose . Consider the expression

where we sum over all partitions with The coefficient is given by the number of matrices with entries either 0 or 1 such that

Here is a sample computation: take the polynomial above and find the coefficient for the monomial For that, we need to find the number of binary matrices with column sums and row sum As expected, we get 5 solutions:

Proof of Theorem.

Using the above example with , we expand and attempt to find the coefficient of E.g. suppose and We wish to expand:

Here is one way we can multiply terms to obtain

Thus each binary matrix corresponds to a way of obtaining by multiplying terms from , , etc. ♦

In the definition of , the variable n is ostensibly missing. However note that if , then the monomial since it would involve more than n variables. For example, if and , then the above expansion gives since One can verify this directly by expanding

On the other hand, if , we now have and now

Exercise

1. Express as a linear combination of the monomial symmetric polynomials: